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On Representations of Orders Over Dedekind Domains

  • D. G. Higman (a1)

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We study representations of o-orders , that is, of o-regular -algebras, in the case that o is a Dedekind domain. Our main concern is with those -modules, called -representation modules, which are regular as o-modules. For any -module M we denote by D(M) the ideal consisting of the elements x ∈ o such that x.Ext1 (M, N) = 0 for all -modules N, where Ext = Ext(,0) is the relative functor of Hochschild (5). To compute D(M) we need the small amount of homological algebra presented in § 1. In § 2 we show that the -representation modules with rational hulls isomorphic to direct sums of right ideal components of the rational hull A of , called principal -modules, are characterized by the property that D(M) ≠ 0. The (, o)-projective -modules are those with D(M) = 0. We observe that D(M) divides the ideal I() of (2) for every M , and give another proof of the fact that I() ≠ 0 if and only if A is separable. Up to this point, o can be taken to be an arbitrary integral domain.

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References

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1. Cartan, H. and Eilenberg, S., Homological algebra (Princeton, 1956).
2. Higman, D.G., On orders in separable algebras, Can. J. Math., 7 (1955), 509515.
3. Cartan, H. and Eilenberg, S. Relative cohomology, Can. J. Math., 9 (1957), 1934.
4. Higman, D.G. and MacLaughlin, J.E., Finiteness of class numbers of representations of algebras over function fields, to appear in the Michigan Journal of Mathematics.
5. Hochschild, G., Relative homological algebra, Trans. Amer. Math. Soc, 82 (1956), 246269.
6. Maranda, J-M., On p-adic integral representations of finite groups, Can. J. Math., 5 (1953), 344355.
7. Maranda, J-M. On the equivalence of representations of finite groups by groups of automorphisms of modules over Dedekind rings, Can. J. Math., 7 (1955), 516526.
8. Reiner, I., Maschke modules over Dekedind rings, Can. J. Math., 8 (1956), 329334.
9. Reiner, I. On class numbers of representations of an order, A. M. S. Notices, 5 (1958), Abstract no. 548-142, 584.
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On Representations of Orders Over Dedekind Domains

  • D. G. Higman (a1)

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